# PosteriorDiffusion ### *class* deepinv.sampling.PosteriorDiffusion(data_fidelity=None, denoiser=None, sde=None, solver=None, dtype=torch.float64, device=torch.device('cpu'), verbose=False, minus_one_one=True, \*args, \*\*kwargs) Bases: [`Reconstructor`](https://deepinv.org/api/stubs/deepinv.models.Reconstructor.html.md#deepinv.models.Reconstructor) Posterior distribution sampling for inverse problems using diffusion models by Reverse-time Stochastic Differential Equation (SDE). Consider the acquisition model: $$ y = \noise{\forw{x}}. $$ This class defines the reverse-time SDE for the posterior distribution $p(x|y)$ given the data $y$: $$ d\, x_t = \left( f(x_t, t) - \frac{1 + \alpha(t)}{2} g(t)^2 \nabla_{x_t} \log p_t(x_t | y) \right) d\,t + g(t) \sqrt{\alpha(t)} d\, w_{t} $$ where $f$ is the drift term, $g$ is the diffusion coefficient and $w$ is the standard Brownian motion. The drift term and the diffusion coefficient are defined by the underlying (unconditional) forward-time SDE `sde`. The (conditional) score function $\nabla_{x_t} \log p_t(x_t | y)$ can be decomposed using the Bayes’ rule: $$ \nabla_{x_t} \log p_t(x_t | y) = \nabla_{x_t} \log p_t(x_t) + \nabla_{x_t} \log p_t(y | x_t). $$ The first term is the score function of the unconditional SDE, which is typically approximated by a MMSE denoiser using the well-known Tweedie’s formula, while the second term is approximated by the (noisy) data-fidelity term. We implement various data-fidelity terms in [`deepinv.sampling.NoisyDataFidelity`](https://deepinv.org/api/stubs/deepinv.sampling.NoisyDataFidelity.html.md#deepinv.sampling.NoisyDataFidelity). * **Parameters:** * **data_fidelity** ([*deepinv.sampling.NoisyDataFidelity*](https://deepinv.org/api/stubs/deepinv.sampling.NoisyDataFidelity.html.md#deepinv.sampling.NoisyDataFidelity)) – the noisy data-fidelity term, used to approximate the score $\nabla_{x_t} \log p_t(y \vert x_t)$. Default to [`deepinv.optim.ZeroFidelity`](https://deepinv.org/api/stubs/deepinv.optim.ZeroFidelity.html.md#deepinv.optim.ZeroFidelity), which corresponds to the zero data-fidelity term and the sampling process boils down to the unconditional SDE sampling. * **denoiser** ([*deepinv.models.Denoiser*](https://deepinv.org/api/stubs/deepinv.models.Denoiser.html.md#deepinv.models.Denoiser)) – a denoiser used to provide an approximation of the (unconditional) score at time $t$ $\nabla \log p_t$. * **sde** ([*deepinv.sampling.DiffusionSDE*](https://deepinv.org/api/stubs/deepinv.sampling.DiffusionSDE.html.md#deepinv.sampling.DiffusionSDE)) – the forward-time SDE, which defines the drift and diffusion terms of the reverse-time SDE. * **solver** ([*deepinv.sampling.BaseSDESolver*](https://deepinv.org/api/stubs/deepinv.sampling.BaseSDESolver.html.md#deepinv.sampling.BaseSDESolver)) – the solver for the SDE. If not specified, the solver from the `sde` will be used. * **dtype** ([*torch.dtype*](https://docs.pytorch.org/docs/stable/tensor_attributes.html#torch.dtype)) – the data type of the sampling solver, except for the `denoiser` which will use `torch.float32`. We recommend using `torch.float64` for better stability and less numerical error when solving the SDE in discrete time, since most computation cost is from evaluating the `denoiser`, which will be always computed in `torch.float32`. * **device** ([*torch.device*](https://docs.pytorch.org/docs/stable/tensor_attributes.html#torch.device)) – the device for the computations. * **verbose** ([*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – whether to display a progress bar during the sampling process, optional. Default to `False`. * **minus_one_one** ([*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – If `True`, wrap the denoiser so that SDE states `x` in `[-1, 1]` are converted to `[0, 1]` before denoising and mapped back afterward. - Set `True` for denoisers trained on `[0, 1]` data range (all denoisers in [`deepinv.models.Denoiser`](https://deepinv.org/api/stubs/deepinv.models.Denoiser.html.md#deepinv.models.Denoiser)). - Set `False` only if the denoiser natively expects `[-1, 1]` data range. This affects only the denoiser interface and usually improves quality when matched to the denoiser’s training range. Default: `True`. #### forward(y, physics, x_init=None, seed=None, timesteps=None, denoise_output=True, get_trajectory=False, \*args, \*\*kwargs) Sample the posterior distribution $p(x|y)$ given the data measurement $y$. * **Parameters:** * **y** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – the data measurement. * **physics** ([*deepinv.physics.Physics*](https://deepinv.org/api/stubs/deepinv.physics.Physics.html.md#deepinv.physics.Physics)) – the forward operator. * **x_init** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) *,* [*tuple*](https://docs.python.org/3.12/library/stdtypes.html#tuple)) – the initial value for the sampling, can be a [`torch.Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) or a tuple `(B, C, H, W)`, indicating the shape of the initial point, matching the shape of `physics` and `y`. In this case, the initial value is taken randomly following the distribution of the `sde` at the first time step of the solver. * **seed** ([*int*](https://docs.python.org/3.12/library/functions.html#int)) – the random seed for reproducibility, the same samples will be generated for the same seed. Default to `None`. * **timesteps** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – the time steps for the solver. If `None`, the default time steps in the solver will be used. Default to `None`. * **denoise_output** ([*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – whether to perform an additional denoising step at the end of the sampling process, which can improve the quality of the generated samples. Default to `True`. * **get_trajectory** ([*bool*](https://docs.python.org/3.12/library/functions.html#bool)) – whether to return the full trajectory of the SDE or only the last sample, optional. Default to `False`. * **\*args** – the additional arguments for the solver. * **\*\*kwargs** – the additional keyword arguments for the solver. * **Returns:** the generated sample ([`torch.Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) of shape `(B, C, H, W)`) if `get_trajectory` is `False`. Otherwise, returns a tuple ([`torch.Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor), [`torch.Tensor`](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) of shape `(B, C, H, W)` and `(N, B, C, H, W)` where `N` is the number of steps. #### score(y, physics, x, t, \*args, \*\*kwargs) Approximating the conditional score $\nabla_{x_t} \log p_t(x_t \vert y)$. * **Parameters:** * **y** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – the data measurement. * **physics** ([*deepinv.physics.Physics*](https://deepinv.org/api/stubs/deepinv.physics.Physics.html.md#deepinv.physics.Physics)) – the forward operator. * **x** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor)) – the current state. * **t** ([*torch.Tensor*](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) *,* [*float*](https://docs.python.org/3.12/library/functions.html#float)) – the current time step. * **\*args** – additional arguments for the score function of the unconditional SDE. * **\*\*kwargs** – additional keyword arguments for the score function of the unconditional SDE. * **Returns:** the score function $\nabla_{x_t} \log p_t(x_t \vert y)$. * **Return type:** [torch.Tensor](https://docs.pytorch.org/docs/stable/tensors.html#torch.Tensor) ## Examples using `PosteriorDiffusion`:
![](auto_examples/sampling/images/thumb/sphx_glr_demo_diffusers_thumb.png) [Using state-of-the-art diffusion models from HuggingFace Diffusers with DeepInverse](https://deepinv.org/auto_examples/sampling/demo_diffusers.html.md)
Using state-of-the-art diffusion models from HuggingFace Diffusers with DeepInverse
![](auto_examples/sampling/images/thumb/sphx_glr_demo_diffusion_sde_thumb.png) [Building your diffusion posterior sampling method using SDEs](https://deepinv.org/auto_examples/sampling/demo_diffusion_sde.html.md)
Building your diffusion posterior sampling method using SDEs
![](auto_examples/sampling/images/thumb/sphx_glr_demo_dps_thumb.png) [DPS – Posterior Sampling with Diffusion Models](https://deepinv.org/auto_examples/sampling/demo_dps.html.md)
DPS -- Posterior Sampling with Diffusion Models
![](auto_examples/sampling/images/thumb/sphx_glr_demo_flow_matching_thumb.png) [Flow-Matching for posterior sampling and unconditional generation](https://deepinv.org/auto_examples/sampling/demo_flow_matching.html.md)
Flow-Matching for posterior sampling and unconditional generation
![](auto_examples/sampling/images/thumb/sphx_glr_demo_noisy_data_fidelity_thumb.png) [Noisy data-fidelity terms for diffusion posterior sampling](https://deepinv.org/auto_examples/sampling/demo_noisy_data_fidelity.html.md)
Noisy data-fidelity terms for diffusion posterior sampling